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          <h1 class="post-title" itemprop="name headline">What is an RKHS [1]</h1>
        

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        <p>RKHS随处可见，本系列仅仅为Dino Sejdinovic, Arthur Gretto的<em>What is an RKHS?</em>整理(个人解读+补充)。<br>第一部分介绍一些<strong>必要的</strong>泛函的概念。本文中的希尔伯特空间定义使用实数和复数，但是举例仅讨论实数。<br><a id="more"></a></p>
<h2 id="Some-functional-analysis-泛函基础"><a href="#Some-functional-analysis-泛函基础" class="headerlink" title="Some functional analysis 泛函基础"></a>Some functional analysis 泛函基础</h2><p>介绍基础泛函概念主要是为了得到两个有用的结论:</p>
<blockquote>
<p>1) 在Banach空间上的线性算子是连续的(continus)$\iff$这个算子是有界的(bounded)<br>2) 所有定义在Banach空间上的连续线性泛函，都来源于内积。(Riesz representation theorem，里斯表示定理)</p>
</blockquote>
<p>这两个结论将帮助我们学习RKHS的性质</p>
<h3 id="Vector-Space-Linear-Space-向量空间-线性空间"><a href="#Vector-Space-Linear-Space-向量空间-线性空间" class="headerlink" title="Vector Space (Linear Space) 向量空间(线性空间)"></a>Vector Space (Linear Space) 向量空间(线性空间)</h3><p>向量空间$\mathcal{V}$是一个在<strong>vector addition</strong>和<strong>scalar multiplication</strong>下闭合的<strong>集合</strong>。比如n-维欧式空间$\mathbb{R}^n$，每个元素就是一个n个实数的序列，标量域就是实数域$\mathbb{R}$。</p>
<p>对于一般的向量空间，标量是域$\mathcal{F}$的成员，那么$\mathcal{V}$就是一个在$\mathcal{F}$上的向量空间。</p>
<p>$\mathcal{V}$要成为在$\mathcal{F}$上的向量空间，$\forall X,Y,Z\in \mathcal{V}$，$\forall r,s\in \mathcal{F}$，必须满足：</p>
<ul>
<li>Commutativity<script type="math/tex; mode=display">X+Y=Y+X</script></li>
<li>Associativity of vector addition<script type="math/tex; mode=display">(X+Y)+Z=X+(Y+Z)</script></li>
<li>Additive identivity <script type="math/tex; mode=display">\forall X, 0+X=X+0=X</script></li>
<li>Existance of additive inverse <script type="math/tex; mode=display">\forall X, \exists -X \space s.t.\space X+(-X)=0</script></li>
<li>Associativity of scalar mutiplication<script type="math/tex; mode=display">r(sX)=(rs)X</script></li>
<li>Distributivity of scalar sums<script type="math/tex; mode=display">(r+s)X=rX+sX</script></li>
<li>Distributivity of vector sums<script type="math/tex; mode=display">r(X+Y)=rX+rY</script></li>
<li>Scalar multiplication identity<script type="math/tex; mode=display">1X=X</script></li>
</ul>
<p>从固定集合$\Omega$到域F的连续实函数集合$\mathcal{f}:\Omega\rightarrow\mathcal{F}$也可以构成向量空间，$(f+g)(w)=f(w)+g(w),(c\cdot f)(x)=c\cdot f(x)$。</p>
<blockquote>
<p>函数空间是一个拓扑向量空间，只不过其中”点”是函数(无限维向量)。<br>齐次微分方程的解就是一个函数空间。所有实函数构成的空间为函数空间$\mathbb{F}$。<br>所有$n$次多项式的集合$\mathbb{P}_n:a_0+a_1x^2+…+a_nx^n$是$\mathbb{F}$的子空间。</p>
</blockquote>
<h3 id="Metric-Space-度量空间-距离空间"><a href="#Metric-Space-度量空间-距离空间" class="headerlink" title="Metric Space 度量空间(距离空间)"></a>Metric Space 度量空间(距离空间)</h3><p>设$\mathcal{X}$为一个<strong>集合</strong>，一个映射$d:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$。$\forall x,y,z\in \mathcal{X}$，有</p>
<ul>
<li>$d(x,y)\ge 0,d(x,y)=0\text{ iff } x=y$</li>
<li>$d(x,y)=d(y,x)$</li>
<li>$d(x,z)\leq d(x,y)+d(y,z)$</li>
</ul>
<p>那么$d$是$\mathcal{X}$的一个度量。称偶对$(\mathcal{X},d)$是一个度量空间。称$\mathcal{X}$为一个对于度量$d$而言的度量空间。</p>
<h3 id="Convergent-sequence-收敛序列"><a href="#Convergent-sequence-收敛序列" class="headerlink" title="Convergent sequence 收敛序列"></a>Convergent sequence 收敛序列</h3><p>一个度量空间$(\mathcal{X},d)$的元素构成的序列$\{x_n\}_{n=1}^\infty$收敛于$x\in\mathcal{X}$。必须满足：</p>
<script type="math/tex; mode=display">\forall x\in\mathcal{X}, \lim_{n\rightarrow\infty}d(x_n,x)=0</script><h3 id="Cauchy-sequence-柯西序列"><a href="#Cauchy-sequence-柯西序列" class="headerlink" title="Cauchy sequence 柯西序列"></a>Cauchy sequence 柯西序列</h3><p>一个度量空间$(\mathcal{X},d)$的元素构成的序列$\{x_n\}_{n=1}^\infty$是柯西序列。必须满足：</p>
<script type="math/tex; mode=display">\lim_{m,n\rightarrow\infty}d(x_n,x_m)=0</script><p>由三角不等式可推出收敛序列都是柯西序列，但是柯西序列不一定是收敛序列。柯西序列都是有界的。</p>
<h3 id="Norm-范数"><a href="#Norm-范数" class="headerlink" title="Norm 范数"></a>Norm 范数</h3><p>$\mathcal{F}$是在$\mathbb{R}$上的向量空间。函数$\parallel\cdot\parallel_\mathcal{F}:\mathcal{F}\rightarrow[0,\infty)$被称作范数需满足：</p>
<ul>
<li>norm separates points<script type="math/tex; mode=display">\parallel f \parallel_\mathcal{F}=0\text{ iff }f = 0</script></li>
<li>positive homogeneity<script type="math/tex; mode=display">\parallel\lambda f \parallel_\mathcal{F}=|\lambda|\parallel f \parallel_\mathcal{F}\forall\lambda\in\mathbb{R},\forall f\in\mathcal{F}</script></li>
<li>triangle inequality<script type="math/tex; mode=display">\parallel f+g \parallel_\mathcal{F}\leq\parallel f \parallel_\mathcal{F}+\parallel g \parallel_\mathcal{F},\forall f,g\in\mathcal{F}</script></li>
</ul>
<p>赋范空间的所有元素必须有有限的范数。如果有一个元素有无限的范数，那它就不在这个空间里。</p>
<blockquote>
<p>经典的赋范空间：$L^p$，$L^\infty$，$l^p$，$c$($c_0$)，$V[a,b]$($V_0[a,b]$)</p>
</blockquote>
<p>范数可以诱导一种在$\mathcal{F}$上的度量：$\mathcal{F}:d(f,g)=\parallel f-g \parallel_\mathcal{F}$。说明$\mathcal{F}$具有某种特定拓扑结构，可以让我们研究连续性和收敛性。</p>
<blockquote>
<p>以绝对值$|\cdot|$为范数的有理数集合$\mathbb{Q}$是一个在其自身上的赋范向量空间， 序列$1, 1.4, 1.41, 1.414, 1.4142, …$是一个$\mathbb{Q}$里的不收敛的柯西序列（因为$\sqrt{2}\notin\mathbb{Q}$）</p>
</blockquote>
<h3 id="Compelete-space-完备空间"><a href="#Compelete-space-完备空间" class="headerlink" title="Compelete space 完备空间"></a>Compelete space 完备空间</h3><p>如果$\mathcal{X}$中的所有柯西序列都收敛(有极限，且极限在$\mathcal{X}$中)，那么$\mathcal{X}$是一个完备空间。</p>
<h3 id="Banach-space-巴拿赫空间"><a href="#Banach-space-巴拿赫空间" class="headerlink" title="Banach space 巴拿赫空间"></a>Banach space 巴拿赫空间</h3><p>巴拿赫空间是一个<strong>完备</strong>的<strong>赋范</strong>空间(包含了自身所有柯西序列的极限)</p>
<h3 id="Inner-product-内积"><a href="#Inner-product-内积" class="headerlink" title="Inner product 内积"></a>Inner product 内积</h3><p>$\mathcal{F}$是一个在$\mathbb{R}$上的向量空间，函数$\langle\cdot,\cdot\rangle:\mathcal{F}\times\mathcal{F}\rightarrow\mathbb{R}$是$\mathcal{F}$上的内积，需满足:</p>
<ul>
<li>$\langle\alpha_1f_1+\alpha_2f_2,g\rangle_\mathcal{F}:\alpha_1\langle f_1,g\rangle_\mathcal{F}+\alpha_2\langle f_2,g\rangle_\mathcal{F}$</li>
<li>$\langle f,g\rangle_\mathcal{F}=\langle g,f\rangle_\mathcal{F}$</li>
<li>$\langle f,f\rangle_\mathcal{F}\geq 0 \text{ and }\langle f,f\rangle_\mathcal{F}=0 \text{ iff }f=0$</li>
</ul>
<p>可以从内积诱导一个范数$\parallel f\parallel_\mathcal{F}=\langle f,f\rangle_\mathcal{F}^{1/2}$</p>
<p>内积与范数有一些实用的性质:</p>
<ul>
<li>Cauchy-Schwarz inequality <script type="math/tex; mode=display">|\langle f,g\rangle|\leq\parallel f\parallel\cdot\parallel g\parallel</script></li>
<li>the parallelogram law<script type="math/tex; mode=display">\parallel f+g\parallel^2+\parallel f-g\parallel^2=2\parallel f\parallel^2+2\parallel g\parallel^2</script></li>
<li>the polarization identity(real)<script type="math/tex; mode=display">4\langle f,g\rangle=\parallel f+g\parallel^2-\parallel f-g\parallel^2</script></li>
</ul>
<blockquote>
<p>空间的关系：内积空间$\subset$赋范空间$\subset$度量空间$\subset$线性空间</p>
</blockquote>
<h3 id="Hilbert-space-希尔伯特空间"><a href="#Hilbert-space-希尔伯特空间" class="headerlink" title="Hilbert space 希尔伯特空间"></a>Hilbert space 希尔伯特空间</h3><p>希尔伯特空间是一个<strong>完备</strong>的<strong>内积</strong>空间，i.e. 有内积的巴拿赫空间。</p>
<blockquote>
<p>向量点积作为内积的$\mathbb{R}^n$是一个有限维的希尔伯特空间<br>内积定义为$\langle f,g\rangle=\int_{-\infty}^{\infty}f(x)g(x)dx$，$\int_{-\infty}^{\infty}f(x)^2dx&lt;\infty$的所有$f:\mathbb{R}\rightarrow\mathbb{R}$的集合，是一个无限维的希尔伯特空间</p>
</blockquote>
<h2 id="Bounded-Continuous-linear-Operators-有界-连续线性算子"><a href="#Bounded-Continuous-linear-Operators-有界-连续线性算子" class="headerlink" title="Bounded/Continuous linear Operators 有界/连续线性算子"></a>Bounded/Continuous linear Operators 有界/连续线性算子</h2><p>以下定义$\mathcal{F}$和$\mathcal{G}$为在$\mathbb{R}$上的赋范线性空间。可以是$f:\mathcal{X}\subset\mathbb{R}\rightarrow\mathbb{R}$，定义$L_p-norm$的Banach空间。</p>
<h3 id="Linear-operator-线性算子"><a href="#Linear-operator-线性算子" class="headerlink" title="Linear operator 线性算子"></a>Linear operator 线性算子</h3><p>$\mathcal{F},\mathcal{G}$是在$\mathbb{R}$上的赋范线性空间。函数$A:\mathcal{F}\rightarrow\mathcal{G}$是线性算子，iff:</p>
<ul>
<li>Homogeneity:<script type="math/tex; mode=display">A(\alpha f)=\alpha(Af),\forall\alpha\in\mathbb{R},f\in\mathcal{F}</script></li>
<li>Additivity:<script type="math/tex; mode=display">A(f+g)=Af+Ag,\forall f,g\in\mathcal{F}</script></li>
</ul>
<blockquote>
<p>$\mathcal{F}$是一个内积空间，$g\in\mathcal{F}$，定义$A_g:\mathcal{F}\rightarrow\mathbb{R}$，$A_g(f):=\langle f,g\rangle_\mathcal{F}$是线性算子。这种从向量空间到标量的映射称为<strong>泛函</strong>。</p>
</blockquote>
<h3 id="Continuity-连续性"><a href="#Continuity-连续性" class="headerlink" title="Continuity 连续性"></a>Continuity 连续性</h3><p>函数$A:\mathcal{F}\rightarrow\mathcal{G}$在$f_0$连续：</p>
<script type="math/tex; mode=display">\forall\epsilon>0,\exists\sigma=\sigma(\epsilon,f_0)>0\text{ s.t. }\parallel f-f_0\parallel\Rightarrow\parallel Af-Af_0\parallel_\mathcal{G}<\epsilon</script><p>如果$\sigma$只取决于$\epsilon$，那么$A$是<strong>一致连续</strong></p>
<h3 id="Lipschitz-continuity-利普希茨连续"><a href="#Lipschitz-continuity-利普希茨连续" class="headerlink" title="Lipschitz continuity 利普希茨连续"></a>Lipschitz continuity 利普希茨连续</h3><p>函数$A:\mathcal{F}\rightarrow\mathcal{G}$在$f_0$利普希茨连续:</p>
<script type="math/tex; mode=display">\exists C>0,\text{ s.t. }\forall f_1,f_2\in\mathcal{F},\parallel Af_1-Af_2\parallel_\mathcal{G}\leq C\parallel f_1-f_2\parallel_\mathcal{f}</script><p>利普希茨连续的函数一致连续，$\sigma=\epsilon/C$</p>
<h3 id="Operator-norm-算子范数"><a href="#Operator-norm-算子范数" class="headerlink" title="Operator norm 算子范数"></a>Operator norm 算子范数</h3><p>线性算子$A:\mathcal{F}\rightarrow\mathcal{G}$的算子范数:</p>
<script type="math/tex; mode=display">\parallel A\parallel=\sup_{f\in\mathcal{F}}\frac{\parallel Af\parallel_\mathcal{G}}{\parallel f\parallel_\mathcal{F}}</script><p>有界算子$\parallel A\parallel&lt;\infty$</p>
<h3 id="Algebraic-dual-代数对偶"><a href="#Algebraic-dual-代数对偶" class="headerlink" title="Algebraic dual 代数对偶"></a>Algebraic dual 代数对偶</h3><p>$\mathcal{F}$是一个赋范空间。<strong>线性</strong>泛函$A:\mathcal{F}\rightarrow\mathbb{R}$的空间$\mathcal{F’}$称为$\mathcal{F}$的代数对偶空间</p>
<h3 id="Topological-dual-拓扑对偶"><a href="#Topological-dual-拓扑对偶" class="headerlink" title="Topological dual 拓扑对偶"></a>Topological dual 拓扑对偶</h3><p>$\mathcal{F}$是一个赋范空间。<strong>连续线性</strong>泛函$A:\mathcal{F}\rightarrow\mathbb{R}$的空间$\mathcal{F’}$称为$\mathcal{F}$的拓扑对偶空间</p>
<h3 id="Riesz-representation-里斯表示定理"><a href="#Riesz-representation-里斯表示定理" class="headerlink" title="Riesz representation 里斯表示定理"></a>Riesz representation 里斯表示定理</h3><p>在希尔伯特空间$\mathcal{F}$，所有<strong>连续线性</strong>泛函具有形式</p>
<script type="math/tex; mode=display">\langle\cdot,g\rangle_\mathcal{F}\text{ for some }g\in\mathcal{F}</script><h3 id="Hilbert-space-isomorphism-希尔伯特空间的同构性"><a href="#Hilbert-space-isomorphism-希尔伯特空间的同构性" class="headerlink" title="Hilbert space isomorphism 希尔伯特空间的同构性"></a>Hilbert space isomorphism 希尔伯特空间的同构性</h3><p>希尔伯特空间$\mathcal{H},\mathcal{F}$是等距同构的:<br>存在<strong>线性双射</strong>$U:\mathcal{H}\rightarrow\mathcal{F}$，同时保持内积$\langle h_1,h_2\rangle_\mathcal{H}=\langle Uh_1,Uh_2\rangle_\mathcal{F}$。</p>

      
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#Some-functional-analysis-泛函基础"><span class="nav-number">1.</span> <span class="nav-text">Some functional analysis 泛函基础</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#Vector-Space-Linear-Space-向量空间-线性空间"><span class="nav-number">1.1.</span> <span class="nav-text">Vector Space (Linear Space) 向量空间(线性空间)</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Metric-Space-度量空间-距离空间"><span class="nav-number">1.2.</span> <span class="nav-text">Metric Space 度量空间(距离空间)</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Convergent-sequence-收敛序列"><span class="nav-number">1.3.</span> <span class="nav-text">Convergent sequence 收敛序列</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Cauchy-sequence-柯西序列"><span class="nav-number">1.4.</span> <span class="nav-text">Cauchy sequence 柯西序列</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Norm-范数"><span class="nav-number">1.5.</span> <span class="nav-text">Norm 范数</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Compelete-space-完备空间"><span class="nav-number">1.6.</span> <span class="nav-text">Compelete space 完备空间</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Banach-space-巴拿赫空间"><span class="nav-number">1.7.</span> <span class="nav-text">Banach space 巴拿赫空间</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Inner-product-内积"><span class="nav-number">1.8.</span> <span class="nav-text">Inner product 内积</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Hilbert-space-希尔伯特空间"><span class="nav-number">1.9.</span> <span class="nav-text">Hilbert space 希尔伯特空间</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Bounded-Continuous-linear-Operators-有界-连续线性算子"><span class="nav-number">2.</span> <span class="nav-text">Bounded/Continuous linear Operators 有界/连续线性算子</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#Linear-operator-线性算子"><span class="nav-number">2.1.</span> <span class="nav-text">Linear operator 线性算子</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Continuity-连续性"><span class="nav-number">2.2.</span> <span class="nav-text">Continuity 连续性</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Lipschitz-continuity-利普希茨连续"><span class="nav-number">2.3.</span> <span class="nav-text">Lipschitz continuity 利普希茨连续</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Operator-norm-算子范数"><span class="nav-number">2.4.</span> <span class="nav-text">Operator norm 算子范数</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Algebraic-dual-代数对偶"><span class="nav-number">2.5.</span> <span class="nav-text">Algebraic dual 代数对偶</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Topological-dual-拓扑对偶"><span class="nav-number">2.6.</span> <span class="nav-text">Topological dual 拓扑对偶</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Riesz-representation-里斯表示定理"><span class="nav-number">2.7.</span> <span class="nav-text">Riesz representation 里斯表示定理</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#Hilbert-space-isomorphism-希尔伯特空间的同构性"><span class="nav-number">2.8.</span> <span class="nav-text">Hilbert space isomorphism 希尔伯特空间的同构性</span></a></li></ol></li></ol></div>
            

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